### Summary of the project

The primary goal of the project is to obtain an
understanding of geometric and dynamical properties of
moduli spaces and mapping class groups.

For the mapping class
group of a surface of finite type, we are interested in subgroups,
in particular in the trace fields of Veech groups
beyond the case of genus 2.

Convex cocompact surface subgroups
are word hyperbolic surface-by-surface groups,
and we aim at clarifying whether or not such groups exist.
Fine asymptotics of the distribution of periodic
orbits for the Teichmüller flow on strata of quadratic
or abelian differentials can be related to dynamical zeta functions.

A Borel conjugacy of the Teichmüller flow on the
moduli space of quadratic differentials into
the Weil-Petersson flow will be used to analyze
dynamical properties of the Weil-Petersson flow.

The handlebody group is a finitely presented
subgroup of the mapping class group which however
is not quasi-isometrically embedded.

A new geometric model for the group
will be used towards obtaining a comprehensive
understanding of the geometry of this group,
in particular with respect to calculating
the Dehn function and quasi-isometric rigidity.

A similar geometric model for the outer automorphism group of a
free group may yield hyperbolicity of the electrified
sphere graph on which this group acts by simplicial automorphisms.

Team members:

Hyungryul Baik

Dawid Kielak

Robert Kucharczyk

Justin Malestein

Mark Pedron

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